LMFDB - Embedded newform 445.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.55334288995$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.737640$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.45589 q^{2} -2.19353 q^{3} +0.119606 q^{4} -1.00000 q^{5} -3.19353 q^{6} +3.71135 q^{7} -2.73764 q^{8} +1.81156 q^{9} +O(q^{10})$	\(q+1.45589 q^{2} -2.19353 q^{3} +0.119606 q^{4} -1.00000 q^{5} -3.19353 q^{6} +3.71135 q^{7} -2.73764 q^{8} +1.81156 q^{9} -1.45589 q^{10} -1.80647 q^{11} -0.262360 q^{12} -6.78527 q^{13} +5.40330 q^{14} +2.19353 q^{15} -4.22491 q^{16} -1.92293 q^{17} +2.63743 q^{18} -6.21801 q^{19} -0.119606 q^{20} -8.14094 q^{21} -2.63002 q^{22} +1.39822 q^{23} +6.00509 q^{24} +1.00000 q^{25} -9.87858 q^{26} +2.60687 q^{27} +0.443901 q^{28} -4.99310 q^{29} +3.19353 q^{30} +1.92427 q^{31} -0.675706 q^{32} +3.96255 q^{33} -2.79958 q^{34} -3.71135 q^{35} +0.216674 q^{36} +2.70709 q^{37} -9.05272 q^{38} +14.8837 q^{39} +2.73764 q^{40} +2.91604 q^{41} -11.8523 q^{42} +0.211586 q^{43} -0.216066 q^{44} -1.81156 q^{45} +2.03564 q^{46} -9.61114 q^{47} +9.26745 q^{48} +6.77411 q^{49} +1.45589 q^{50} +4.21801 q^{51} -0.811561 q^{52} +10.9331 q^{53} +3.79531 q^{54} +1.80647 q^{55} -10.1603 q^{56} +13.6394 q^{57} -7.26939 q^{58} -12.0984 q^{59} +0.262360 q^{60} +10.1997 q^{61} +2.80152 q^{62} +6.72333 q^{63} +7.46606 q^{64} +6.78527 q^{65} +5.76902 q^{66} +2.88548 q^{67} -0.229995 q^{68} -3.06702 q^{69} -5.40330 q^{70} -3.85919 q^{71} -4.95940 q^{72} +11.5899 q^{73} +3.94121 q^{74} -2.19353 q^{75} -0.743713 q^{76} -6.70445 q^{77} +21.6689 q^{78} -10.8158 q^{79} +4.22491 q^{80} -11.1529 q^{81} +4.24542 q^{82} -5.59257 q^{83} -0.973708 q^{84} +1.92293 q^{85} +0.308045 q^{86} +10.9525 q^{87} +4.94547 q^{88} -1.00000 q^{89} -2.63743 q^{90} -25.1825 q^{91} +0.167235 q^{92} -4.22094 q^{93} -13.9927 q^{94} +6.21801 q^{95} +1.48218 q^{96} +1.07064 q^{97} +9.86233 q^{98} -3.27254 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9} - q^{10} - 14 q^{11} - 3 q^{12} - 5 q^{13} - 5 q^{14} + 2 q^{15} - 3 q^{16} - 3 q^{17} + 7 q^{18} - q^{19} - 3 q^{20} - 4 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 4 q^{25} - 9 q^{26} + q^{27} + 5 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 9 q^{36} + 3 q^{37} + 2 q^{38} + 11 q^{39} + 9 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} - 9 q^{44} + 4 q^{45} - 4 q^{46} - 24 q^{47} + 21 q^{48} + q^{50} - 7 q^{51} + 8 q^{52} + 3 q^{53} + 17 q^{54} + 14 q^{55} - 13 q^{56} + 19 q^{57} + 19 q^{58} - 22 q^{59} + 3 q^{60} - 3 q^{61} - 24 q^{62} + 6 q^{63} - 11 q^{64} + 5 q^{65} + 14 q^{66} - 9 q^{67} + 31 q^{68} + 7 q^{69} + 5 q^{70} + 16 q^{71} + 10 q^{72} + 3 q^{73} + 31 q^{74} - 2 q^{75} - 24 q^{76} - 4 q^{77} + 16 q^{78} - 27 q^{79} + 3 q^{80} - 8 q^{81} - 15 q^{82} + 6 q^{83} + 7 q^{84} + 3 q^{85} + 15 q^{86} + 4 q^{87} + 30 q^{88} - 4 q^{89} - 7 q^{90} - 29 q^{91} - 17 q^{92} - 2 q^{93} + 13 q^{94} + q^{95} + 12 q^{96} + 41 q^{97} + 22 q^{98} + 21 q^{99}+O(q^{100}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9 - q^10 - 14 * q^11 - 3 * q^12 - 5 * q^13 - 5 * q^14 + 2 * q^15 - 3 * q^16 - 3 * q^17 + 7 * q^18 - q^19 - 3 * q^20 - 4 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 4 * q^25 - 9 * q^26 + q^27 + 5 * q^28 - 10 * q^29 + 6 * q^30 - 11 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 9 * q^36 + 3 * q^37 + 2 * q^38 + 11 * q^39 + 9 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 - 9 * q^44 + 4 * q^45 - 4 * q^46 - 24 * q^47 + 21 * q^48 + q^50 - 7 * q^51 + 8 * q^52 + 3 * q^53 + 17 * q^54 + 14 * q^55 - 13 * q^56 + 19 * q^57 + 19 * q^58 - 22 * q^59 + 3 * q^60 - 3 * q^61 - 24 * q^62 + 6 * q^63 - 11 * q^64 + 5 * q^65 + 14 * q^66 - 9 * q^67 + 31 * q^68 + 7 * q^69 + 5 * q^70 + 16 * q^71 + 10 * q^72 + 3 * q^73 + 31 * q^74 - 2 * q^75 - 24 * q^76 - 4 * q^77 + 16 * q^78 - 27 * q^79 + 3 * q^80 - 8 * q^81 - 15 * q^82 + 6 * q^83 + 7 * q^84 + 3 * q^85 + 15 * q^86 + 4 * q^87 + 30 * q^88 - 4 * q^89 - 7 * q^90 - 29 * q^91 - 17 * q^92 - 2 * q^93 + 13 * q^94 + q^95 + 12 * q^96 + 41 * q^97 + 22 * q^98 + 21 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.45589	1.02947	0.514734	−	0.857350i	$-0.327891\pi$
$2$	1.45589	1.02947	0.514734	+	0.857350i	$0.327891\pi$
$3$	−2.19353	−1.26643	−0.633217	−	0.773975i	$-0.718266\pi$
$3$	−2.19353	−1.26643	−0.633217	+	0.773975i	$0.718266\pi$
$4$	0.119606	0.0598032
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.19353	−1.30375
$7$	3.71135	1.40276	0.701379	−	0.712789i	$-0.252568\pi$
$7$	3.71135	1.40276	0.701379	+	0.712789i	$0.252568\pi$
$8$	−2.73764	−0.967902
$9$	1.81156	0.603854
$10$	−1.45589	−0.460392
$11$	−1.80647	−0.544672	−0.272336	−	0.962202i	$-0.587796\pi$
$11$	−1.80647	−0.544672	−0.272336	+	0.962202i	$0.587796\pi$
$12$	−0.262360	−0.0757367
$13$	−6.78527	−1.88190	−0.940948	−	0.338552i	$-0.890063\pi$
$13$	−6.78527	−1.88190	−0.940948	+	0.338552i	$0.890063\pi$
$14$	5.40330	1.44409
$15$	2.19353	0.566366
$16$	−4.22491	−1.05623
$17$	−1.92293	−0.466380	−0.233190	−	0.972431i	$-0.574916\pi$
$17$	−1.92293	−0.466380	−0.233190	+	0.972431i	$0.574916\pi$
$18$	2.63743	0.621648
$19$	−6.21801	−1.42651	−0.713255	−	0.700905i	$-0.752780\pi$
$19$	−6.21801	−1.42651	−0.713255	+	0.700905i	$0.752780\pi$
$20$	−0.119606	−0.0267448
$21$	−8.14094	−1.77650
$22$	−2.63002	−0.560722
$23$	1.39822	0.291548	0.145774	−	0.989318i	$-0.453433\pi$
$23$	1.39822	0.291548	0.145774	+	0.989318i	$0.453433\pi$
$24$	6.00509	1.22578
$25$	1.00000	0.200000
$26$	−9.87858	−1.93735
$27$	2.60687	0.501693
$28$	0.443901	0.0838894
$29$	−4.99310	−0.927196	−0.463598	−	0.886046i	$-0.653442\pi$
$29$	−4.99310	−0.927196	−0.463598	+	0.886046i	$0.653442\pi$
$30$	3.19353	0.583056
$31$	1.92427	0.345609	0.172805	−	0.984956i	$-0.444717\pi$
$31$	1.92427	0.345609	0.172805	+	0.984956i	$0.444717\pi$
$32$	−0.675706	−0.119449
$33$	3.96255	0.689791
$34$	−2.79958	−0.480123
$35$	−3.71135	−0.627332
$36$	0.216674	0.0361124
$37$	2.70709	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$37$	2.70709	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$38$	−9.05272	−1.46854
$39$	14.8837	2.38329
$40$	2.73764	0.432859
$41$	2.91604	0.455408	0.227704	−	0.973730i	$-0.426878\pi$
$41$	2.91604	0.455408	0.227704	+	0.973730i	$0.426878\pi$
$42$	−11.8523	−1.82885
$43$	0.211586	0.0322666	0.0161333	−	0.999870i	$-0.494864\pi$
$43$	0.211586	0.0322666	0.0161333	+	0.999870i	$0.494864\pi$
$44$	−0.216066	−0.0325731
$45$	−1.81156	−0.270052
$46$	2.03564	0.300139
$47$	−9.61114	−1.40193	−0.700964	−	0.713197i	$-0.747247\pi$
$47$	−9.61114	−1.40193	−0.700964	+	0.713197i	$0.747247\pi$
$48$	9.26745	1.33764
$49$	6.77411	0.967730
$50$	1.45589	0.205893
$51$	4.21801	0.590639
$52$	−0.811561	−0.112543
$53$	10.9331	1.50178	0.750889	−	0.660428i	$-0.229625\pi$
$53$	10.9331	1.50178	0.750889	+	0.660428i	$0.229625\pi$
$54$	3.79531	0.516477
$55$	1.80647	0.243585
$56$	−10.1603	−1.35773
$57$	13.6394	1.80658
$58$	−7.26939	−0.954518
$59$	−12.0984	−1.57508	−0.787539	−	0.616265i	$-0.788645\pi$
$59$	−12.0984	−1.57508	−0.787539	+	0.616265i	$0.788645\pi$
$60$	0.262360	0.0338705
$61$	10.1997	1.30594	0.652971	−	0.757383i	$-0.273522\pi$
$61$	10.1997	1.30594	0.652971	+	0.757383i	$0.273522\pi$
$62$	2.80152	0.355793
$63$	6.72333	0.847061
$64$	7.46606	0.933258
$65$	6.78527	0.841609
$66$	5.76902	0.710117
$67$	2.88548	0.352518	0.176259	−	0.984344i	$-0.443600\pi$
$67$	2.88548	0.352518	0.176259	+	0.984344i	$0.443600\pi$
$68$	−0.229995	−0.0278910
$69$	−3.06702	−0.369226
$70$	−5.40330	−0.645818
$71$	−3.85919	−0.458002	−0.229001	−	0.973426i	$-0.573546\pi$
$71$	−3.85919	−0.458002	−0.229001	+	0.973426i	$0.573546\pi$
$72$	−4.95940	−0.584471
$73$	11.5899	1.35650	0.678250	−	0.734832i	$-0.262739\pi$
$73$	11.5899	1.35650	0.678250	+	0.734832i	$0.262739\pi$
$74$	3.94121	0.458156
$75$	−2.19353	−0.253287
$76$	−0.743713	−0.0853098
$77$	−6.70445	−0.764043
$78$	21.6689	2.45352
$79$	−10.8158	−1.21688	−0.608438	−	0.793602i	$-0.708203\pi$
$79$	−10.8158	−1.21688	−0.608438	+	0.793602i	$0.708203\pi$
$80$	4.22491	0.472359
$81$	−11.1529	−1.23921
$82$	4.24542	0.468828
$83$	−5.59257	−0.613864	−0.306932	−	0.951731i	$-0.599302\pi$
$83$	−5.59257	−0.613864	−0.306932	+	0.951731i	$0.599302\pi$
$84$	−0.973708	−0.106240
$85$	1.92293	0.208572
$86$	0.308045	0.0332174
$87$	10.9525	1.17423
$88$	4.94547	0.527189
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	−2.63743	−0.278009
$91$	−25.1825	−2.63984
$92$	0.167235	0.0174355
$93$	−4.22094	−0.437691
$94$	−13.9927	−1.44324
$95$	6.21801	0.637954
$96$	1.48218	0.151274
$97$	1.07064	0.108707	0.0543536	−	0.998522i	$-0.482690\pi$
$97$	1.07064	0.108707	0.0543536	+	0.998522i	$0.482690\pi$
$98$	9.86233	0.996246
$99$	−3.27254	−0.328902
$100$	0.119606	0.0119606
$101$	−13.8603	−1.37915	−0.689576	−	0.724213i	$-0.742203\pi$
$101$	−13.8603	−1.37915	−0.689576	+	0.724213i	$0.742203\pi$
$102$	6.14094	0.608044
$103$	2.85216	0.281032	0.140516	−	0.990078i	$-0.455124\pi$
$103$	2.85216	0.281032	0.140516	+	0.990078i	$0.455124\pi$
$104$	18.5756	1.82149
$105$	8.14094	0.794475
$106$	15.9174	1.54603
$107$	−1.48908	−0.143954	−0.0719772	−	0.997406i	$-0.522931\pi$
$107$	−1.48908	−0.143954	−0.0719772	+	0.997406i	$0.522931\pi$
$108$	0.311798	0.0300028
$109$	9.40330	0.900673	0.450337	−	0.892859i	$-0.351304\pi$
$109$	9.40330	0.900673	0.450337	+	0.892859i	$0.351304\pi$
$110$	2.63002	0.250763
$111$	−5.93807	−0.563616
$112$	−15.6801	−1.48163
$113$	12.6841	1.19322	0.596609	−	0.802532i	$-0.296515\pi$
$113$	12.6841	1.19322	0.596609	+	0.802532i	$0.296515\pi$
$114$	19.8574	1.85981
$115$	−1.39822	−0.130384
$116$	−0.597207	−0.0554492
$117$	−12.2919	−1.13639
$118$	−17.6139	−1.62149
$119$	−7.13668	−0.654218
$120$	−6.00509	−0.548187
$121$	−7.73666	−0.703332
$122$	14.8497	1.34442
$123$	−6.39641	−0.576744
$124$	0.230155	0.0206685
$125$	−1.00000	−0.0894427
$126$	9.78841	0.872021
$127$	1.25413	0.111286	0.0556429	−	0.998451i	$-0.482279\pi$
$127$	1.25413	0.111286	0.0556429	+	0.998451i	$0.482279\pi$
$128$	12.2212	1.08021
$129$	−0.464120	−0.0408635
$130$	9.87858	0.866409
$131$	−2.27240	−0.198541	−0.0992704	−	0.995060i	$-0.531651\pi$
$131$	−2.27240	−0.198541	−0.0992704	+	0.995060i	$0.531651\pi$
$132$	0.473946	0.0412517
$133$	−23.0772	−2.00105
$134$	4.20093	0.362905
$135$	−2.60687	−0.224364
$136$	5.26430	0.451410
$137$	−15.5836	−1.33140	−0.665700	−	0.746219i	$-0.731867\pi$
$137$	−15.5836	−1.33140	−0.665700	+	0.746219i	$0.731867\pi$
$138$	−4.46524	−0.380106
$139$	−21.5004	−1.82364	−0.911819	−	0.410593i	$-0.865322\pi$
$139$	−21.5004	−1.82364	−0.911819	+	0.410593i	$0.865322\pi$
$140$	−0.443901	−0.0375165
$141$	21.0823	1.77545
$142$	−5.61854	−0.471498
$143$	12.2574	1.02502
$144$	−7.65368	−0.637806
$145$	4.99310	0.414655
$146$	16.8736	1.39647
$147$	−14.8592	−1.22557
$148$	0.323785	0.0266149
$149$	−17.0028	−1.39292	−0.696461	−	0.717595i	$-0.745243\pi$
$149$	−17.0028	−1.39292	−0.696461	+	0.717595i	$0.745243\pi$
$150$	−3.19353	−0.260750
$151$	−17.7866	−1.44745	−0.723727	−	0.690087i	$-0.757572\pi$
$151$	−17.7866	−1.44745	−0.723727	+	0.690087i	$0.757572\pi$
$152$	17.0227	1.38072
$153$	−3.48351	−0.281625
$154$	−9.76092	−0.786557
$155$	−1.92427	−0.154561
$156$	1.78018	0.142529
$157$	−8.13061	−0.648893	−0.324447	−	0.945904i	$-0.605178\pi$
$157$	−8.13061	−0.648893	−0.324447	+	0.945904i	$0.605178\pi$
$158$	−15.7466	−1.25273
$159$	−23.9821	−1.90190
$160$	0.675706	0.0534192
$161$	5.18926	0.408971
$162$	−16.2374	−1.27573
$163$	23.0135	1.80255	0.901276	−	0.433244i	$-0.142631\pi$
$163$	23.0135	1.80255	0.901276	+	0.433244i	$0.142631\pi$
$164$	0.348776	0.0272349
$165$	−3.96255	−0.308484
$166$	−8.14214	−0.631953
$167$	5.88974	0.455762	0.227881	−	0.973689i	$-0.426820\pi$
$167$	5.88974	0.455762	0.227881	+	0.973689i	$0.426820\pi$
$168$	22.2870	1.71948
$169$	33.0399	2.54153
$170$	2.79958	0.214718
$171$	−11.2643	−0.861403
$172$	0.0253070	0.00192964
$173$	−15.0723	−1.14593	−0.572964	−	0.819581i	$-0.694206\pi$
$173$	−15.0723	−1.14593	−0.572964	+	0.819581i	$0.694206\pi$
$174$	15.9456	1.20883
$175$	3.71135	0.280552
$176$	7.63218	0.575297
$177$	26.5382	1.99473
$178$	−1.45589	−0.109123
$179$	4.65562	0.347977	0.173989	−	0.984748i	$-0.444334\pi$
$179$	4.65562	0.347977	0.173989	+	0.984748i	$0.444334\pi$
$180$	−0.216674	−0.0161499
$181$	9.62070	0.715101	0.357551	−	0.933894i	$-0.383612\pi$
$181$	9.62070	0.715101	0.357551	+	0.933894i	$0.383612\pi$
$182$	−36.6629	−2.71763
$183$	−22.3734	−1.65389
$184$	−3.82781	−0.282190
$185$	−2.70709	−0.199029
$186$	−6.14521	−0.450588
$187$	3.47373	0.254024
$188$	−1.14955	−0.0838397
$189$	9.67501	0.703754
$190$	9.05272	0.656753
$191$	13.3629	0.966907	0.483454	−	0.875370i	$-0.339382\pi$
$191$	13.3629	0.966907	0.483454	+	0.875370i	$0.339382\pi$
$192$	−16.3770	−1.18191
$193$	0.449680	0.0323687	0.0161843	−	0.999869i	$-0.494848\pi$
$193$	0.449680	0.0323687	0.0161843	+	0.999869i	$0.494848\pi$
$194$	1.55873	0.111911
$195$	−14.8837	−1.06584
$196$	0.810226	0.0578733
$197$	8.56942	0.610546	0.305273	−	0.952265i	$-0.401252\pi$
$197$	8.56942	0.610546	0.305273	+	0.952265i	$0.401252\pi$
$198$	−4.76444	−0.338594
$199$	−8.67704	−0.615099	−0.307550	−	0.951532i	$-0.599509\pi$
$199$	−8.67704	−0.615099	−0.307550	+	0.951532i	$0.599509\pi$
$200$	−2.73764	−0.193580
$201$	−6.32938	−0.446440
$202$	−20.1790	−1.41979
$203$	−18.5311	−1.30063
$204$	0.504501	0.0353221
$205$	−2.91604	−0.203665
$206$	4.15242	0.289313
$207$	2.53295	0.176052
$208$	28.6671	1.98771
$209$	11.2327	0.776980
$210$	11.8523	0.817886
$211$	25.1953	1.73452	0.867258	−	0.497858i	$-0.165880\pi$
$211$	25.1953	1.73452	0.867258	+	0.497858i	$0.165880\pi$
$212$	1.30767	0.0898111
$213$	8.46524	0.580029
$214$	−2.16793	−0.148196
$215$	−0.211586	−0.0144300
$216$	−7.13668	−0.485590
$217$	7.14163	0.484806
$218$	13.6901	0.927214
$219$	−25.4228	−1.71792
$220$	0.216066	0.0145671
$221$	13.0476	0.877679
$222$	−8.64515	−0.580225
$223$	16.2438	1.08777	0.543884	−	0.839161i	$-0.316953\pi$
$223$	16.2438	1.08777	0.543884	+	0.839161i	$0.316953\pi$
$224$	−2.50778	−0.167558
$225$	1.81156	0.120771
$226$	18.4666	1.22838
$227$	−13.1941	−0.875726	−0.437863	−	0.899042i	$-0.644264\pi$
$227$	−13.1941	−0.875726	−0.437863	+	0.899042i	$0.644264\pi$
$228$	1.63135	0.108039
$229$	2.36998	0.156613	0.0783063	−	0.996929i	$-0.475049\pi$
$229$	2.36998	0.156613	0.0783063	+	0.996929i	$0.475049\pi$
$230$	−2.03564	−0.134226
$231$	14.7064	0.967610
$232$	13.6693	0.897435
$233$	−9.67877	−0.634077	−0.317039	−	0.948413i	$-0.602688\pi$
$233$	−9.67877	−0.634077	−0.317039	+	0.948413i	$0.602688\pi$
$234$	−17.8957	−1.16988
$235$	9.61114	0.626961
$236$	−1.44705	−0.0941946
$237$	23.7248	1.54109
$238$	−10.3902	−0.673497
$239$	−23.8871	−1.54513	−0.772564	−	0.634936i	$-0.781026\pi$
$239$	−23.8871	−1.54513	−0.772564	+	0.634936i	$0.781026\pi$
$240$	−9.26745	−0.598211
$241$	−15.7221	−1.01275	−0.506376	−	0.862313i	$-0.669015\pi$
$241$	−15.7221	−1.01275	−0.506376	+	0.862313i	$0.669015\pi$
$242$	−11.2637	−0.724058
$243$	16.6436	1.06769
$244$	1.21995	0.0780995
$245$	−6.77411	−0.432782
$246$	−9.31244	−0.593740
$247$	42.1909	2.68454
$248$	−5.26796	−0.334516
$249$	12.2674	0.777418
$250$	−1.45589	−0.0920784
$251$	14.2933	0.902183	0.451091	−	0.892478i	$-0.351035\pi$
$251$	14.2933	0.902183	0.451091	+	0.892478i	$0.351035\pi$
$252$	0.804153	0.0506569
$253$	−2.52584	−0.158798
$254$	1.82587	0.114565
$255$	−4.21801	−0.264142
$256$	2.86049	0.178781
$257$	−8.51369	−0.531070	−0.265535	−	0.964101i	$-0.585549\pi$
$257$	−8.51369	−0.531070	−0.265535	+	0.964101i	$0.585549\pi$
$258$	−0.675706	−0.0420676
$259$	10.0469	0.624286
$260$	0.811561	0.0503309
$261$	−9.04531	−0.559891
$262$	−3.30836	−0.204391
$263$	−19.3754	−1.19474	−0.597370	−	0.801966i	$-0.703788\pi$
$263$	−19.3754	−1.19474	−0.597370	+	0.801966i	$0.703788\pi$
$264$	−10.8480	−0.667650
$265$	−10.9331	−0.671616
$266$	−33.5978	−2.06001
$267$	2.19353	0.134242
$268$	0.345122	0.0210817
$269$	−7.39853	−0.451096	−0.225548	−	0.974232i	$-0.572417\pi$
$269$	−7.39853	−0.451096	−0.225548	+	0.974232i	$0.572417\pi$
$270$	−3.79531	−0.230975
$271$	−27.2477	−1.65518	−0.827590	−	0.561333i	$-0.810289\pi$
$271$	−27.2477	−1.65518	−0.827590	+	0.561333i	$0.810289\pi$
$272$	8.12422	0.492603
$273$	55.2385	3.34319
$274$	−22.6880	−1.37063
$275$	−1.80647	−0.108934
$276$	−0.366835	−0.0220809
$277$	28.4582	1.70989	0.854943	−	0.518722i	$-0.173592\pi$
$277$	28.4582	1.70989	0.854943	+	0.518722i	$0.173592\pi$
$278$	−31.3021	−1.87738
$279$	3.48593	0.208697
$280$	10.1603	0.607196
$281$	22.1815	1.32323	0.661617	−	0.749842i	$-0.269870\pi$
$281$	22.1815	1.32323	0.661617	+	0.749842i	$0.269870\pi$
$282$	30.6934	1.82777
$283$	0.247445	0.0147091	0.00735455	−	0.999973i	$-0.497659\pi$
$283$	0.247445	0.0147091	0.00735455	+	0.999973i	$0.497659\pi$
$284$	−0.461584	−0.0273899
$285$	−13.6394	−0.807927
$286$	17.8454	1.05522
$287$	10.8224	0.638828
$288$	−1.22408	−0.0721297
$289$	−13.3023	−0.782490
$290$	7.26939	0.426873
$291$	−2.34848	−0.137671
$292$	1.38623	0.0811229
$293$	−11.3844	−0.665085	−0.332542	−	0.943088i	$-0.607906\pi$
$293$	−11.3844	−0.665085	−0.332542	+	0.943088i	$0.607906\pi$
$294$	−21.6333	−1.26168
$295$	12.0984	0.704396
$296$	−7.41103	−0.430757
$297$	−4.70925	−0.273258
$298$	−24.7541	−1.43397
$299$	−9.48727	−0.548663
$300$	−0.262360	−0.0151473
$301$	0.785269	0.0452622
$302$	−25.8953	−1.49011
$303$	30.4030	1.74660
$304$	26.2705	1.50672
$305$	−10.1997	−0.584035
$306$	−5.07160	−0.289924
$307$	−6.37110	−0.363618	−0.181809	−	0.983334i	$-0.558195\pi$
$307$	−6.37110	−0.363618	−0.181809	+	0.983334i	$0.558195\pi$
$308$	−0.801895	−0.0456922
$309$	−6.25629	−0.355908
$310$	−2.80152	−0.159116
$311$	−30.5996	−1.73514	−0.867572	−	0.497312i	$-0.834321\pi$
$311$	−30.5996	−1.73514	−0.867572	+	0.497312i	$0.834321\pi$
$312$	−40.7461	−2.30680
$313$	−19.2316	−1.08704	−0.543518	−	0.839398i	$-0.682908\pi$
$313$	−19.2316	−1.08704	−0.543518	+	0.839398i	$0.682908\pi$
$314$	−11.8372	−0.668014
$315$	−6.72333	−0.378817
$316$	−1.29364	−0.0727730
$317$	−9.70429	−0.545047	−0.272524	−	0.962149i	$-0.587858\pi$
$317$	−9.70429	−0.545047	−0.272524	+	0.962149i	$0.587858\pi$
$318$	−34.9152	−1.95795
$319$	9.01990	0.505018
$320$	−7.46606	−0.417366
$321$	3.26633	0.182309
$322$	7.55498	0.421023
$323$	11.9568	0.665296
$324$	−1.33396	−0.0741089
$325$	−6.78527	−0.376379
$326$	33.5050	1.85567
$327$	−20.6264	−1.14064
$328$	−7.98306	−0.440791
$329$	−35.6703	−1.96657
$330$	−5.76902	−0.317574
$331$	−28.9275	−1.59000	−0.795001	−	0.606608i	$-0.792530\pi$
$331$	−28.9275	−1.59000	−0.795001	+	0.606608i	$0.792530\pi$
$332$	−0.668906	−0.0367110
$333$	4.90405	0.268740
$334$	8.57480	0.469192
$335$	−2.88548	−0.157651
$336$	34.3947	1.87639
$337$	2.30865	0.125760	0.0628802	−	0.998021i	$-0.479971\pi$
$337$	2.30865	0.125760	0.0628802	+	0.998021i	$0.479971\pi$
$338$	48.1023	2.61642
$339$	−27.8229	−1.51113
$340$	0.229995	0.0124732
$341$	−3.47614	−0.188244
$342$	−16.3996	−0.886786
$343$	−0.838363	−0.0452673
$344$	−0.579246	−0.0312309
$345$	3.06702	0.165123
$346$	−21.9436	−1.17970
$347$	−34.4649	−1.85017	−0.925085	−	0.379759i	$-0.876007\pi$
$347$	−34.4649	−1.85017	−0.925085	+	0.379759i	$0.876007\pi$
$348$	1.30999	0.0702228
$349$	19.3154	1.03393	0.516965	−	0.856007i	$-0.327062\pi$
$349$	19.3154	1.03393	0.516965	+	0.856007i	$0.327062\pi$
$350$	5.40330	0.288819
$351$	−17.6883	−0.944133
$352$	1.22064	0.0650605
$353$	1.41183	0.0751441	0.0375721	−	0.999294i	$-0.488038\pi$
$353$	1.41183	0.0751441	0.0375721	+	0.999294i	$0.488038\pi$
$354$	38.6366	2.05351
$355$	3.85919	0.204825
$356$	−0.119606	−0.00633912
$357$	15.6545	0.828524
$358$	6.77806	0.358231
$359$	−17.2200	−0.908834	−0.454417	−	0.890789i	$-0.650152\pi$
$359$	−17.2200	−0.908834	−0.454417	+	0.890789i	$0.650152\pi$
$360$	4.95940	0.261383
$361$	19.6636	1.03493
$362$	14.0067	0.736173
$363$	16.9706	0.890724
$364$	−3.01199	−0.157871
$365$	−11.5899	−0.606645
$366$	−32.5731	−1.70262
$367$	−20.4193	−1.06588	−0.532939	−	0.846154i	$-0.678913\pi$
$367$	−20.4193	−1.06588	−0.532939	+	0.846154i	$0.678913\pi$
$368$	−5.90733	−0.307941
$369$	5.28258	0.275000
$370$	−3.94121	−0.204894
$371$	40.5766	2.10663
$372$	−0.504851	−0.0261753
$373$	−0.00410346	−0.000212469 0	−0.000106235	−	1.00000i	$-0.500034\pi$
$373$	−0.00410346	−0.000212469 0	−0.000106235		1.00000i	$0.500034\pi$
$374$	5.05736	0.261510
$375$	2.19353	0.113273
$376$	26.3118	1.35693
$377$	33.8795	1.74489
$378$	14.0857	0.724492
$379$	33.3456	1.71285	0.856424	−	0.516273i	$-0.172681\pi$
$379$	33.3456	1.71285	0.856424	+	0.516273i	$0.172681\pi$
$380$	0.743713	0.0381517
$381$	−2.75096	−0.140936
$382$	19.4549	0.995399
$383$	34.6473	1.77039	0.885196	−	0.465218i	$-0.154024\pi$
$383$	34.6473	1.77039	0.885196	+	0.465218i	$0.154024\pi$
$384$	−26.8074	−1.36801
$385$	6.70445	0.341690
$386$	0.654683	0.0333225
$387$	0.383301	0.0194843
$388$	0.128056	0.00650104
$389$	30.7954	1.56139	0.780696	−	0.624912i	$-0.214865\pi$
$389$	30.7954	1.56139	0.780696	+	0.624912i	$0.214865\pi$
$390$	−21.6689	−1.09725
$391$	−2.68868	−0.135972
$392$	−18.5451	−0.936668
$393$	4.98458	0.251439
$394$	12.4761	0.628537
$395$	10.8158	0.544203
$396$	−0.391416	−0.0196694
$397$	30.1664	1.51401	0.757004	−	0.653411i	$-0.226662\pi$
$397$	30.1664	1.51401	0.757004	+	0.653411i	$0.226662\pi$
$398$	−12.6328	−0.633224
$399$	50.6205	2.53419
$400$	−4.22491	−0.211245
$401$	−7.51947	−0.375504	−0.187752	−	0.982216i	$-0.560120\pi$
$401$	−7.51947	−0.375504	−0.187752	+	0.982216i	$0.560120\pi$
$402$	−9.21486	−0.459596
$403$	−13.0567	−0.650400
$404$	−1.65778	−0.0824777
$405$	11.1529	0.554194
$406$	−26.9792	−1.33896
$407$	−4.89028	−0.242402
$408$	−11.5474	−0.571681
$409$	32.8936	1.62648	0.813241	−	0.581927i	$-0.197701\pi$
$409$	32.8936	1.62648	0.813241	+	0.581927i	$0.197701\pi$
$410$	−4.24542	−0.209666
$411$	34.1831	1.68613
$412$	0.341136	0.0168066
$413$	−44.9014	−2.20945
$414$	3.68769	0.181240
$415$	5.59257	0.274528
$416$	4.58484	0.224790
$417$	47.1616	2.30952
$418$	16.3535	0.799875
$419$	15.0851	0.736957	0.368479	−	0.929636i	$-0.379879\pi$
$419$	15.0851	0.736957	0.368479	+	0.929636i	$0.379879\pi$
$420$	0.973708	0.0475121
$421$	−23.0575	−1.12375	−0.561876	−	0.827221i	$-0.689920\pi$
$421$	−23.0575	−1.12375	−0.561876	+	0.827221i	$0.689920\pi$
$422$	36.6815	1.78563
$423$	−17.4112	−0.846560
$424$	−29.9309	−1.45357
$425$	−1.92293	−0.0932760
$426$	12.3244	0.597120
$427$	37.8548	1.83192
$428$	−0.178103	−0.00860893
$429$	−26.8870	−1.29811
$430$	−0.308045	−0.0148553
$431$	−12.8238	−0.617703	−0.308851	−	0.951110i	$-0.599945\pi$
$431$	−12.8238	−0.617703	−0.308851	+	0.951110i	$0.599945\pi$
$432$	−11.0138	−0.529901
$433$	−28.7231	−1.38034	−0.690171	−	0.723646i	$-0.742465\pi$
$433$	−28.7231	−1.38034	−0.690171	+	0.723646i	$0.742465\pi$
$434$	10.3974	0.499092
$435$	−10.9525	−0.525132
$436$	1.12469	0.0538631
$437$	−8.69411	−0.415896
$438$	−37.0128	−1.76854
$439$	−23.6489	−1.12870	−0.564349	−	0.825536i	$-0.690873\pi$
$439$	−23.6489	−1.12870	−0.564349	+	0.825536i	$0.690873\pi$
$440$	−4.94547	−0.235766
$441$	12.2717	0.584367
$442$	18.9959	0.903541
$443$	−35.5043	−1.68686	−0.843430	−	0.537239i	$-0.819467\pi$
$443$	−35.5043	−1.68686	−0.843430	+	0.537239i	$0.819467\pi$
$444$	−0.710230	−0.0337060
$445$	1.00000	0.0474045
$446$	23.6492	1.11982
$447$	37.2960	1.76404
$448$	27.7092	1.30913
$449$	32.7779	1.54688	0.773442	−	0.633867i	$-0.218533\pi$
$449$	32.7779	1.54688	0.773442	+	0.633867i	$0.218533\pi$
$450$	2.63743	0.124330
$451$	−5.26774	−0.248048
$452$	1.51710	0.0713582
$453$	39.0154	1.83310
$454$	−19.2092	−0.901531
$455$	25.1825	1.18057
$456$	−37.3397	−1.74859
$457$	28.7506	1.34490	0.672448	−	0.740145i	$-0.265243\pi$
$457$	28.7506	1.34490	0.672448	+	0.740145i	$0.265243\pi$
$458$	3.45042	0.161228
$459$	−5.01285	−0.233980
$460$	−0.167235	−0.00779739
$461$	−11.6097	−0.540717	−0.270358	−	0.962760i	$-0.587142\pi$
$461$	−11.6097	−0.540717	−0.270358	+	0.962760i	$0.587142\pi$
$462$	21.4108	0.996123
$463$	−18.7448	−0.871145	−0.435573	−	0.900154i	$-0.643454\pi$
$463$	−18.7448	−0.871145	−0.435573	+	0.900154i	$0.643454\pi$
$464$	21.0954	0.979329
$465$	4.22094	0.195741
$466$	−14.0912	−0.652762
$467$	−0.634016	−0.0293387	−0.0146694	−	0.999892i	$-0.504670\pi$
$467$	−0.634016	−0.0293387	−0.0146694	+	0.999892i	$0.504670\pi$
$468$	−1.47019	−0.0679597
$469$	10.7090	0.494497
$470$	13.9927	0.645436
$471$	17.8347	0.821780
$472$	33.1211	1.52452
$473$	−0.382224	−0.0175747
$474$	34.5406	1.58650
$475$	−6.21801	−0.285302
$476$	−0.853592	−0.0391243
$477$	19.8060	0.906854
$478$	−34.7769	−1.59066
$479$	29.2698	1.33737	0.668686	−	0.743545i	$-0.266857\pi$
$479$	29.2698	1.33737	0.668686	+	0.743545i	$0.266857\pi$
$480$	−1.48218	−0.0676519
$481$	−18.3683	−0.837523
$482$	−22.8896	−1.04259
$483$	−11.3828	−0.517935
$484$	−0.925353	−0.0420615
$485$	−1.07064	−0.0486154
$486$	24.2312	1.09915
$487$	8.96526	0.406255	0.203127	−	0.979152i	$-0.434889\pi$
$487$	8.96526	0.406255	0.203127	+	0.979152i	$0.434889\pi$
$488$	−27.9232	−1.26402
$489$	−50.4806	−2.28281
$490$	−9.86233	−0.445535
$491$	21.4601	0.968479	0.484240	−	0.874935i	$-0.339096\pi$
$491$	21.4601	0.968479	0.484240	+	0.874935i	$0.339096\pi$
$492$	−0.765050	−0.0344911
$493$	9.60141	0.432426
$494$	61.4251	2.76365
$495$	3.27254	0.147090
$496$	−8.12986	−0.365041
$497$	−14.3228	−0.642465
$498$	17.8600	0.800326
$499$	21.8821	0.979576	0.489788	−	0.871841i	$-0.337074\pi$
$499$	21.8821	0.979576	0.489788	+	0.871841i	$0.337074\pi$
$500$	−0.119606	−0.00534896
$501$	−12.9193	−0.577192
$502$	20.8094	0.928768
$503$	−8.91940	−0.397696	−0.198848	−	0.980030i	$-0.563720\pi$
$503$	−8.91940	−0.397696	−0.198848	+	0.980030i	$0.563720\pi$
$504$	−18.4061	−0.819872
$505$	13.8603	0.616776
$506$	−3.67733	−0.163477
$507$	−72.4739	−3.21868
$508$	0.150001	0.00665524
$509$	−5.76572	−0.255561	−0.127780	−	0.991802i	$-0.540785\pi$
$509$	−5.76572	−0.255561	−0.127780	+	0.991802i	$0.540785\pi$
$510$	−6.14094	−0.271926
$511$	43.0143	1.90284
$512$	−20.2778	−0.896159
$513$	−16.2096	−0.715670
$514$	−12.3950	−0.546719
$515$	−2.85216	−0.125681
$516$	−0.0555116	−0.00244376
$517$	17.3623	0.763591
$518$	14.6272	0.642682
$519$	33.0616	1.45124
$520$	−18.5756	−0.814595
$521$	14.8063	0.648676	0.324338	−	0.945941i	$-0.394858\pi$
$521$	14.8063	0.648676	0.324338	+	0.945941i	$0.394858\pi$
$522$	−13.1689	−0.576389
$523$	34.3182	1.50063	0.750315	−	0.661081i	$-0.229902\pi$
$523$	34.3182	1.50063	0.750315	+	0.661081i	$0.229902\pi$
$524$	−0.271794	−0.0118734
$525$	−8.14094	−0.355300
$526$	−28.2084	−1.22995
$527$	−3.70024	−0.161185
$528$	−16.7414	−0.728576
$529$	−21.0450	−0.915000
$530$	−15.9174	−0.691406
$531$	−21.9170	−0.951117
$532$	−2.76018	−0.119669
$533$	−19.7861	−0.857031
$534$	3.19353	0.138197
$535$	1.48908	0.0643784
$536$	−7.89941	−0.341203
$537$	−10.2122	−0.440690
$538$	−10.7714	−0.464389
$539$	−12.2372	−0.527095
$540$	−0.311798	−0.0134177
$541$	−19.0816	−0.820384	−0.410192	−	0.911999i	$-0.634538\pi$
$541$	−19.0816	−0.820384	−0.410192	+	0.911999i	$0.634538\pi$
$542$	−39.6696	−1.70395
$543$	−21.1033	−0.905628
$544$	1.29934	0.0557086
$545$	−9.40330	−0.402793
$546$	80.4210	3.44170
$547$	−16.2497	−0.694789	−0.347394	−	0.937719i	$-0.612933\pi$
$547$	−16.2497	−0.694789	−0.347394	+	0.937719i	$0.612933\pi$
$548$	−1.86390	−0.0796220
$549$	18.4774	0.788598
$550$	−2.63002	−0.112144
$551$	31.0472	1.32265
$552$	8.39641	0.357375
$553$	−40.1413	−1.70698
$554$	41.4319	1.76027
$555$	5.93807	0.252057
$556$	−2.57158	−0.109059
$557$	34.7023	1.47038	0.735191	−	0.677860i	$-0.237092\pi$
$557$	34.7023	1.47038	0.735191	+	0.677860i	$0.237092\pi$
$558$	5.07512	0.214847
$559$	−1.43567	−0.0607223
$560$	15.6801	0.662605
$561$	−7.61972	−0.321705
$562$	32.2937	1.36223
$563$	11.6228	0.489842	0.244921	−	0.969543i	$-0.421238\pi$
$563$	11.6228	0.489842	0.244921	+	0.969543i	$0.421238\pi$
$564$	2.52157	0.106177
$565$	−12.6841	−0.533623
$566$	0.360253	0.0151425
$567$	−41.3924	−1.73832
$568$	10.5651	0.443301
$569$	42.6106	1.78633	0.893164	−	0.449732i	$-0.148480\pi$
$569$	42.6106	1.78633	0.893164	+	0.449732i	$0.148480\pi$
$570$	−19.8574	−0.831734
$571$	−3.41985	−0.143116	−0.0715582	−	0.997436i	$-0.522797\pi$
$571$	−3.41985	−0.143116	−0.0715582	+	0.997436i	$0.522797\pi$
$572$	1.46606	0.0612992
$573$	−29.3119	−1.22452
$574$	15.7562	0.657652
$575$	1.39822	0.0583096
$576$	13.5252	0.563551
$577$	6.58744	0.274239	0.137119	−	0.990555i	$-0.456216\pi$
$577$	6.58744	0.274239	0.137119	+	0.990555i	$0.456216\pi$
$578$	−19.3667	−0.805548
$579$	−0.986386	−0.0409928
$580$	0.597207	0.0247977
$581$	−20.7560	−0.861103
$582$	−3.41913	−0.141727
$583$	−19.7504	−0.817977
$584$	−31.7291	−1.31296
$585$	12.2919	0.508209
$586$	−16.5744	−0.684683
$587$	−3.93893	−0.162577	−0.0812884	−	0.996691i	$-0.525903\pi$
$587$	−3.93893	−0.162577	−0.0812884	+	0.996691i	$0.525903\pi$
$588$	−1.77725	−0.0732927
$589$	−11.9651	−0.493014
$590$	17.6139	0.725153
$591$	−18.7973	−0.773216
$592$	−11.4372	−0.470065
$593$	−20.9067	−0.858537	−0.429269	−	0.903177i	$-0.641229\pi$
$593$	−20.9067	−0.858537	−0.429269	+	0.903177i	$0.641229\pi$
$594$	−6.85613	−0.281310
$595$	7.13668	0.292575
$596$	−2.03364	−0.0833011
$597$	19.0333	0.778982
$598$	−13.8124	−0.564830
$599$	8.80091	0.359595	0.179798	−	0.983704i	$-0.442456\pi$
$599$	8.80091	0.359595	0.179798	+	0.983704i	$0.442456\pi$
$600$	6.00509	0.245157
$601$	3.04731	0.124303	0.0621513	−	0.998067i	$-0.480204\pi$
$601$	3.04731	0.124303	0.0621513	+	0.998067i	$0.480204\pi$
$602$	1.14326	0.0465959
$603$	5.22723	0.212869
$604$	−2.12739	−0.0865623
$605$	7.73666	0.314540
$606$	44.2633	1.79807
$607$	−37.2207	−1.51074	−0.755371	−	0.655298i	$-0.772543\pi$
$607$	−37.2207	−1.51074	−0.755371	+	0.655298i	$0.772543\pi$
$608$	4.20154	0.170395
$609$	40.6486	1.64716
$610$	−14.8497	−0.601245
$611$	65.2141	2.63828
$612$	−0.416650	−0.0168421
$613$	7.03831	0.284275	0.142137	−	0.989847i	$-0.454603\pi$
$613$	7.03831	0.284275	0.142137	+	0.989847i	$0.454603\pi$
$614$	−9.27560	−0.374333
$615$	6.39641	0.257928
$616$	18.3544	0.739519
$617$	−41.9256	−1.68786	−0.843931	−	0.536452i	$-0.819764\pi$
$617$	−41.9256	−1.68786	−0.843931	+	0.536452i	$0.819764\pi$
$618$	−9.10845	−0.366395
$619$	−7.92233	−0.318425	−0.159213	−	0.987244i	$-0.550896\pi$
$619$	−7.92233	−0.318425	−0.159213	+	0.987244i	$0.550896\pi$
$620$	−0.230155	−0.00924324
$621$	3.64497	0.146268
$622$	−44.5496	−1.78627
$623$	−3.71135	−0.148692
$624$	−62.8821	−2.51730
$625$	1.00000	0.0400000
$626$	−27.9991	−1.11907
$627$	−24.6392	−0.983993
$628$	−0.972472	−0.0388059
$629$	−5.20555	−0.207559
$630$	−9.78841	−0.389980
$631$	27.9812	1.11391	0.556956	−	0.830542i	$-0.311969\pi$
$631$	27.9812	1.11391	0.556956	+	0.830542i	$0.311969\pi$
$632$	29.6098	1.17782
$633$	−55.2666	−2.19665
$634$	−14.1283	−0.561108
$635$	−1.25413	−0.0497685
$636$	−2.86841	−0.113740
$637$	−45.9641	−1.82117
$638$	13.1320	0.519899
$639$	−6.99116	−0.276566
$640$	−12.2212	−0.483084
$641$	−10.8328	−0.427868	−0.213934	−	0.976848i	$-0.568628\pi$
$641$	−10.8328	−0.427868	−0.213934	+	0.976848i	$0.568628\pi$
$642$	4.75541	0.187681
$643$	33.1604	1.30772	0.653859	−	0.756616i	$-0.273149\pi$
$643$	33.1604	1.30772	0.653859	+	0.756616i	$0.273149\pi$
$644$	0.620669	0.0244578
$645$	0.464120	0.0182747
$646$	17.4078	0.684900
$647$	7.80677	0.306916	0.153458	−	0.988155i	$-0.450959\pi$
$647$	7.80677	0.306916	0.153458	+	0.988155i	$0.450959\pi$
$648$	30.5327	1.19944
$649$	21.8554	0.857901
$650$	−9.87858	−0.387470
$651$	−15.6654	−0.613974
$652$	2.75255	0.107798
$653$	−9.79178	−0.383182	−0.191591	−	0.981475i	$-0.561365\pi$
$653$	−9.79178	−0.383182	−0.191591	+	0.981475i	$0.561365\pi$
$654$	−30.0297	−1.17425
$655$	2.27240	0.0887901
$656$	−12.3200	−0.481015
$657$	20.9959	0.819127
$658$	−51.9319	−2.02452
$659$	−33.0449	−1.28725	−0.643623	−	0.765342i	$-0.722570\pi$
$659$	−33.0449	−1.28725	−0.643623	+	0.765342i	$0.722570\pi$
$660$	−0.473946	−0.0184483
$661$	21.9348	0.853165	0.426582	−	0.904449i	$-0.359717\pi$
$661$	21.9348	0.853165	0.426582	+	0.904449i	$0.359717\pi$
$662$	−42.1152	−1.63686
$663$	−28.6203	−1.11152
$664$	15.3104	0.594160
$665$	23.0772	0.894895
$666$	7.13974	0.276659
$667$	−6.98143	−0.270322
$668$	0.704451	0.0272560
$669$	−35.6313	−1.37758
$670$	−4.20093	−0.162296
$671$	−18.4255	−0.711310
$672$	5.50088	0.212201
$673$	12.6202	0.486472	0.243236	−	0.969967i	$-0.421791\pi$
$673$	12.6202	0.486472	0.243236	+	0.969967i	$0.421791\pi$
$674$	3.36114	0.129466
$675$	2.60687	0.100339
$676$	3.95178	0.151991
$677$	−43.1296	−1.65761	−0.828803	−	0.559541i	$-0.810977\pi$
$677$	−43.1296	−1.65761	−0.828803	+	0.559541i	$0.810977\pi$
$678$	−40.5069	−1.55566
$679$	3.97353	0.152490
$680$	−5.26430	−0.201877
$681$	28.9417	1.10905
$682$	−5.06087	−0.193791
$683$	−33.1031	−1.26666	−0.633328	−	0.773884i	$-0.718311\pi$
$683$	−33.1031	−1.26666	−0.633328	+	0.773884i	$0.718311\pi$
$684$	−1.34728	−0.0515146
$685$	15.5836	0.595420
$686$	−1.22056	−0.0466012
$687$	−5.19862	−0.198340
$688$	−0.893931	−0.0340808
$689$	−74.1841	−2.82619
$690$	4.46524	0.169989
$691$	11.4778	0.436637	0.218319	−	0.975878i	$-0.429943\pi$
$691$	11.4778	0.436637	0.218319	+	0.975878i	$0.429943\pi$
$692$	−1.80275	−0.0685301
$693$	−12.1455	−0.461370
$694$	−50.1769	−1.90469
$695$	21.5004	0.815556
$696$	−29.9840	−1.13654
$697$	−5.60735	−0.212393
$698$	28.1210	1.06440
$699$	21.2306	0.803017
$700$	0.443901	0.0167779
$701$	27.7156	1.04680	0.523401	−	0.852087i	$-0.324663\pi$
$701$	27.7156	1.04680	0.523401	+	0.852087i	$0.324663\pi$
$702$	−25.7522	−0.971955
$703$	−16.8327	−0.634857
$704$	−13.4872	−0.508320
$705$	−21.0823	−0.794005
$706$	2.05546	0.0773584
$707$	−51.4404	−1.93462
$708$	3.17413	0.119291
$709$	21.6735	0.813965	0.406983	−	0.913436i	$-0.366581\pi$
$709$	21.6735	0.813965	0.406983	+	0.913436i	$0.366581\pi$
$710$	5.61854	0.210860
$711$	−19.5935	−0.734815
$712$	2.73764	0.102597
$713$	2.69054	0.100762
$714$	22.7912	0.852939
$715$	−12.2574	−0.458401
$716$	0.556842	0.0208101
$717$	52.3970	1.95680
$718$	−25.0703	−0.935615
$719$	13.5991	0.507160	0.253580	−	0.967314i	$-0.418392\pi$
$719$	13.5991	0.507160	0.253580	+	0.967314i	$0.418392\pi$
$720$	7.65368	0.285236
$721$	10.5854	0.394219
$722$	28.6280	1.06542
$723$	34.4869	1.28258
$724$	1.15070	0.0427653
$725$	−4.99310	−0.185439
$726$	24.7072	0.916971
$727$	22.5512	0.836379	0.418189	−	0.908360i	$-0.362665\pi$
$727$	22.5512	0.836379	0.418189	+	0.908360i	$0.362665\pi$
$728$	68.9406	2.55511
$729$	−3.04947	−0.112943
$730$	−16.8736	−0.624521
$731$	−0.406866	−0.0150485
$732$	−2.67600	−0.0989078
$733$	11.3716	0.420019	0.210009	−	0.977699i	$-0.432651\pi$
$733$	11.3716	0.420019	0.210009	+	0.977699i	$0.432651\pi$
$734$	−29.7282	−1.09729
$735$	14.8592	0.548089
$736$	−0.944782	−0.0348251
$737$	−5.21254	−0.192007
$738$	7.69084	0.283104
$739$	−51.8437	−1.90710	−0.953551	−	0.301232i	$-0.902602\pi$
$739$	−51.8437	−1.90710	−0.953551	+	0.301232i	$0.902602\pi$
$740$	−0.323785	−0.0119026
$741$	−92.5468	−3.39979
$742$	59.0749	2.16871
$743$	20.9239	0.767623	0.383812	−	0.923411i	$-0.374611\pi$
$743$	20.9239	0.767623	0.383812	+	0.923411i	$0.374611\pi$
$744$	11.5554	0.423642
$745$	17.0028	0.622933
$746$	−0.00597418	−0.000218730 0
$747$	−10.1313	−0.370684
$748$	0.415480	0.0151915
$749$	−5.52648	−0.201933
$750$	3.19353	0.116611
$751$	27.2392	0.993972	0.496986	−	0.867759i	$-0.334440\pi$
$751$	27.2392	0.993972	0.496986	+	0.867759i	$0.334440\pi$
$752$	40.6062	1.48075
$753$	−31.3527	−1.14255
$754$	49.3248	1.79630
$755$	17.7866	0.647321
$756$	1.15719	0.0420867
$757$	−18.6306	−0.677142	−0.338571	−	0.940941i	$-0.609943\pi$
$757$	−18.6306	−0.677142	−0.338571	+	0.940941i	$0.609943\pi$
$758$	48.5474	1.76332
$759$	5.54049	0.201107
$760$	−17.0227	−0.617477
$761$	−4.06378	−0.147312	−0.0736559	−	0.997284i	$-0.523467\pi$
$761$	−4.06378	−0.147312	−0.0736559	+	0.997284i	$0.523467\pi$
$762$	−4.00509	−0.145089
$763$	34.8989	1.26343
$764$	1.59829	0.0578241
$765$	3.48351	0.125947
$766$	50.4425	1.82256
$767$	82.0909	2.96413
$768$	−6.27456	−0.226414
$769$	−32.7396	−1.18062	−0.590310	−	0.807177i	$-0.700994\pi$
$769$	−32.7396	−1.18062	−0.590310	+	0.807177i	$0.700994\pi$
$770$	9.76092	0.351759
$771$	18.6750	0.672564
$772$	0.0537846	0.00193575
$773$	19.0463	0.685049	0.342524	−	0.939509i	$-0.388718\pi$
$773$	19.0463	0.685049	0.342524	+	0.939509i	$0.388718\pi$
$774$	0.558043	0.0200584
$775$	1.92427	0.0691218
$776$	−2.93103	−0.105218
$777$	−22.0382	−0.790617
$778$	44.8347	1.60740
$779$	−18.1319	−0.649644
$780$	−1.78018	−0.0637407
$781$	6.97152	0.249461
$782$	−3.91441	−0.139979
$783$	−13.0164	−0.465168
$784$	−28.6200	−1.02214
$785$	8.13061	0.290194
$786$	7.25698	0.258848
$787$	36.9703	1.31785	0.658924	−	0.752210i	$-0.271012\pi$
$787$	36.9703	1.31785	0.658924	+	0.752210i	$0.271012\pi$
$788$	1.02496	0.0365126
$789$	42.5005	1.51306
$790$	15.7466	0.560239
$791$	47.0750	1.67379
$792$	8.95903	0.318345
$793$	−69.2079	−2.45765
$794$	43.9188	1.55862
$795$	23.9821	0.850557
$796$	−1.03783	−0.0367849
$797$	−10.1404	−0.359191	−0.179596	−	0.983741i	$-0.557479\pi$
$797$	−10.1404	−0.359191	−0.179596	+	0.983741i	$0.557479\pi$
$798$	73.6977	2.60887
$799$	18.4816	0.653832
$800$	−0.675706	−0.0238898
$801$	−1.81156	−0.0640084
$802$	−10.9475	−0.386570
$803$	−20.9369	−0.738847
$804$	−0.757034	−0.0266985
$805$	−5.18926	−0.182898
$806$	−19.0091	−0.669566
$807$	16.2289	0.571284
$808$	37.9445	1.33488
$809$	0.167200	0.00587845	0.00293922	−	0.999996i	$-0.499064\pi$
$809$	0.167200	0.00587845	0.00293922	+	0.999996i	$0.499064\pi$
$810$	16.2374	0.570524
$811$	−41.1102	−1.44357	−0.721787	−	0.692115i	$-0.756679\pi$
$811$	−41.1102	−1.44357	−0.721787	+	0.692115i	$0.756679\pi$
$812$	−2.21644	−0.0777819
$813$	59.7686	2.09618
$814$	−7.11969	−0.249545
$815$	−23.0135	−0.806126
$816$	−17.8207	−0.623849
$817$	−1.31564	−0.0460285
$818$	47.8893	1.67441
$819$	−45.6196	−1.59408
$820$	−0.348776	−0.0121798
$821$	−43.1698	−1.50664	−0.753318	−	0.657656i	$-0.771548\pi$
$821$	−43.1698	−1.50664	−0.753318	+	0.657656i	$0.771548\pi$
$822$	49.7668	1.73582
$823$	−8.46445	−0.295052	−0.147526	−	0.989058i	$-0.547131\pi$
$823$	−8.46445	−0.295052	−0.147526	+	0.989058i	$0.547131\pi$
$824$	−7.80818	−0.272011
$825$	3.96255	0.137958
$826$	−65.3713	−2.27456
$827$	6.41763	0.223163	0.111581	−	0.993755i	$-0.464408\pi$
$827$	6.41763	0.223163	0.111581	+	0.993755i	$0.464408\pi$
$828$	0.302957	0.0105285
$829$	−16.7935	−0.583261	−0.291631	−	0.956531i	$-0.594198\pi$
$829$	−16.7935	−0.583261	−0.291631	+	0.956531i	$0.594198\pi$
$830$	8.14214	0.282618
$831$	−62.4238	−2.16546
$832$	−50.6592	−1.75629
$833$	−13.0262	−0.451330
$834$	68.6620	2.37757
$835$	−5.88974	−0.203823
$836$	1.34350	0.0464658
$837$	5.01633	0.173390
$838$	21.9623	0.758673
$839$	−7.92834	−0.273717	−0.136858	−	0.990591i	$-0.543701\pi$
$839$	−7.92834	−0.273717	−0.136858	+	0.990591i	$0.543701\pi$
$840$	−22.2870	−0.768974
$841$	−4.06893	−0.140308
$842$	−33.5691	−1.15687
$843$	−48.6556	−1.67579
$844$	3.01352	0.103730
$845$	−33.0399	−1.13661
$846$	−25.3487	−0.871505
$847$	−28.7134	−0.986605
$848$	−46.1914	−1.58622
$849$	−0.542778	−0.0186281
$850$	−2.79958	−0.0960246
$851$	3.78509	0.129751
$852$	1.01250	0.0346875
$853$	−27.1727	−0.930375	−0.465187	−	0.885212i	$-0.654013\pi$
$853$	−27.1727	−0.930375	−0.465187	+	0.885212i	$0.654013\pi$
$854$	55.1123	1.88590
$855$	11.2643	0.385231
$856$	4.07656	0.139334
$857$	−43.2642	−1.47788	−0.738938	−	0.673773i	$-0.764672\pi$
$857$	−43.2642	−1.47788	−0.738938	+	0.673773i	$0.764672\pi$
$858$	−39.1444	−1.33637
$859$	18.4487	0.629463	0.314731	−	0.949181i	$-0.398086\pi$
$859$	18.4487	0.629463	0.314731	+	0.949181i	$0.398086\pi$
$860$	−0.0253070	−0.000862962 0
$861$	−23.7393	−0.809033
$862$	−18.6701	−0.635905
$863$	1.54533	0.0526037	0.0263018	−	0.999654i	$-0.491627\pi$
$863$	1.54533	0.0526037	0.0263018	+	0.999654i	$0.491627\pi$
$864$	−1.76148	−0.0599267
$865$	15.0723	0.512475
$866$	−41.8175	−1.42102
$867$	29.1790	0.990971
$868$	0.854185	0.0289929
$869$	19.5385	0.662798
$870$	−15.9456	−0.540607
$871$	−19.5788	−0.663401
$872$	−25.7429	−0.871763
$873$	1.93953	0.0656433
$874$	−12.6576	−0.428151
$875$	−3.71135	−0.125466
$876$	−3.04073	−0.102737
$877$	47.7230	1.61149	0.805745	−	0.592262i	$-0.201765\pi$
$877$	47.7230	1.61149	0.805745	+	0.592262i	$0.201765\pi$
$878$	−34.4301	−1.16196
$879$	24.9720	0.842286
$880$	−7.63218	−0.257281
$881$	28.2004	0.950095	0.475047	−	0.879960i	$-0.342431\pi$
$881$	28.2004	0.950095	0.475047	+	0.879960i	$0.342431\pi$
$882$	17.8662	0.601587
$883$	−11.2269	−0.377816	−0.188908	−	0.981995i	$-0.560495\pi$
$883$	−11.2269	−0.377816	−0.188908	+	0.981995i	$0.560495\pi$
$884$	1.56058	0.0524879
$885$	−26.5382	−0.892071
$886$	−51.6902	−1.73657
$887$	12.8249	0.430619	0.215309	−	0.976546i	$-0.430924\pi$
$887$	12.8249	0.430619	0.215309	+	0.976546i	$0.430924\pi$
$888$	16.2563	0.545525
$889$	4.65450	0.156107
$890$	1.45589	0.0488014
$891$	20.1475	0.674965
$892$	1.94286	0.0650519
$893$	59.7621	1.99986
$894$	54.2988	1.81602
$895$	−4.65562	−0.155620
$896$	45.3570	1.51527
$897$	20.8106	0.694845
$898$	47.7209	1.59247
$899$	−9.60807	−0.320447
$900$	0.216674	0.00722247
$901$	−21.0237	−0.700400
$902$	−7.66923	−0.255358
$903$	−1.72251	−0.0573215
$904$	−34.7244	−1.15492
$905$	−9.62070	−0.319803
$906$	56.8020	1.88712
$907$	−45.9112	−1.52446	−0.762229	−	0.647308i	$-0.775895\pi$
$907$	−45.9112	−1.52446	−0.762229	+	0.647308i	$0.775895\pi$
$908$	−1.57810	−0.0523712
$909$	−25.1088	−0.832806
$910$	36.6629	1.21536
$911$	55.7475	1.84700	0.923499	−	0.383600i	$-0.125316\pi$
$911$	55.7475	1.84700	0.923499	+	0.383600i	$0.125316\pi$
$912$	−57.6251	−1.90816
$913$	10.1028	0.334355
$914$	41.8576	1.38453
$915$	22.3734	0.739642
$916$	0.283465	0.00936593
$917$	−8.43368	−0.278505
$918$	−7.29814	−0.240874
$919$	−49.0675	−1.61859	−0.809293	−	0.587405i	$-0.800150\pi$
$919$	−49.0675	−1.61859	−0.809293	+	0.587405i	$0.800150\pi$
$920$	3.82781	0.126199
$921$	13.9752	0.460498
$922$	−16.9024	−0.556650
$923$	26.1856	0.861911
$924$	1.75898	0.0578661
$925$	2.70709	0.0890084
$926$	−27.2903	−0.896815
$927$	5.16686	0.169702
$928$	3.37387	0.110753
$929$	−0.564626	−0.0185248	−0.00926238	−	0.999957i	$-0.502948\pi$
$929$	−0.564626	−0.0185248	−0.00926238	+	0.999957i	$0.502948\pi$
$930$	6.14521	0.201509
$931$	−42.1215	−1.38048
$932$	−1.15764	−0.0379198
$933$	67.1210	2.19744
$934$	−0.923055	−0.0302033
$935$	−3.47373	−0.113603
$936$	33.6509	1.09991
$937$	21.4710	0.701426	0.350713	−	0.936483i	$-0.385939\pi$
$937$	21.4710	0.701426	0.350713	+	0.936483i	$0.385939\pi$
$938$	15.5911	0.509069
$939$	42.1851	1.37666
$940$	1.14955	0.0374943
$941$	32.9258	1.07335	0.536676	−	0.843789i	$-0.319680\pi$
$941$	32.9258	1.07335	0.536676	+	0.843789i	$0.319680\pi$
$942$	25.9653	0.845996
$943$	4.07725	0.132773
$944$	51.1146	1.66364
$945$	−9.67501	−0.314728
$946$	−0.556475	−0.0180926
$947$	−46.4200	−1.50845	−0.754224	−	0.656617i	$-0.771987\pi$
$947$	−46.4200	−1.50845	−0.754224	+	0.656617i	$0.771987\pi$
$948$	2.83764	0.0921621
$949$	−78.6408	−2.55279
$950$	−9.05272	−0.293709
$951$	21.2866	0.690266
$952$	19.5377	0.633219
$953$	40.6651	1.31727	0.658636	−	0.752462i	$-0.271134\pi$
$953$	40.6651	1.31727	0.658636	+	0.752462i	$0.271134\pi$
$954$	28.8353	0.933577
$955$	−13.3629	−0.432414
$956$	−2.85705	−0.0924036
$957$	−19.7854	−0.639571
$958$	42.6135	1.37678
$959$	−57.8363	−1.86763
$960$	16.3770	0.528566
$961$	−27.2972	−0.880554
$962$	−26.7422	−0.862202
$963$	−2.69755	−0.0869274
$964$	−1.88047	−0.0605657
$965$	−0.449680	−0.0144757
$966$	−16.5721	−0.533197
$967$	13.7463	0.442051	0.221026	−	0.975268i	$-0.429060\pi$
$967$	13.7463	0.442051	0.221026	+	0.975268i	$0.429060\pi$
$968$	21.1802	0.680757
$969$	−26.2276	−0.842553
$970$	−1.55873	−0.0500479
$971$	−10.4871	−0.336548	−0.168274	−	0.985740i	$-0.553819\pi$
$971$	−10.4871	−0.336548	−0.168274	+	0.985740i	$0.553819\pi$
$972$	1.99068	0.0638512
$973$	−79.7954	−2.55812
$974$	13.0524	0.418226
$975$	14.8837	0.476659
$976$	−43.0929	−1.37937
$977$	26.8721	0.859716	0.429858	−	0.902897i	$-0.358564\pi$
$977$	26.8721	0.859716	0.429858	+	0.902897i	$0.358564\pi$
$978$	−73.4941	−2.35008
$979$	1.80647	0.0577351
$980$	−0.810226	−0.0258817
$981$	17.0347	0.543875
$982$	31.2434	0.997018
$983$	5.47408	0.174596	0.0872980	−	0.996182i	$-0.472177\pi$
$983$	5.47408	0.174596	0.0872980	+	0.996182i	$0.472177\pi$
$984$	17.5111	0.558232
$985$	−8.56942	−0.273044
$986$	13.9786	0.445168
$987$	78.2437	2.49052
$988$	5.04629	0.160544
$989$	0.295843	0.00940725
$990$	4.76444	0.151424
$991$	25.1620	0.799296	0.399648	−	0.916669i	$-0.369132\pi$
$991$	25.1620	0.799296	0.399648	+	0.916669i	$0.369132\pi$
$992$	−1.30024	−0.0412827
$993$	63.4534	2.01363
$994$	−20.8524	−0.661397
$995$	8.67704	0.275081
$996$	1.46726	0.0464920
$997$	11.3414	0.359186	0.179593	−	0.983741i	$-0.442522\pi$
$997$	11.3414	0.359186	0.179593	+	0.983741i	$0.442522\pi$
$998$	31.8578	1.00844
$999$	7.05703	0.223274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.d.1.3	✓	4
3.2	odd	2		4005.2.a.l.1.2		4
4.3	odd	2		7120.2.a.bc.1.4		4
5.4	even	2		2225.2.a.i.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.d.1.3	✓	4	1.1	even	1	trivial
2225.2.a.i.1.2		4	5.4	even	2
4005.2.a.l.1.2		4	3.2	odd	2
7120.2.a.bc.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q+1.45589 q^{2} -2.19353 q^{3} +0.119606 q^{4} -1.00000 q^{5} -3.19353 q^{6} +3.71135 q^{7} -2.73764 q^{8} +1.81156 q^{9} +O(q^{10})\)	\(q+1.45589 q^{2} -2.19353 q^{3} +0.119606 q^{4} -1.00000 q^{5} -3.19353 q^{6} +3.71135 q^{7} -2.73764 q^{8} +1.81156 q^{9} -1.45589 q^{10} -1.80647 q^{11} -0.262360 q^{12} -6.78527 q^{13} +5.40330 q^{14} +2.19353 q^{15} -4.22491 q^{16} -1.92293 q^{17} +2.63743 q^{18} -6.21801 q^{19} -0.119606 q^{20} -8.14094 q^{21} -2.63002 q^{22} +1.39822 q^{23} +6.00509 q^{24} +1.00000 q^{25} -9.87858 q^{26} +2.60687 q^{27} +0.443901 q^{28} -4.99310 q^{29} +3.19353 q^{30} +1.92427 q^{31} -0.675706 q^{32} +3.96255 q^{33} -2.79958 q^{34} -3.71135 q^{35} +0.216674 q^{36} +2.70709 q^{37} -9.05272 q^{38} +14.8837 q^{39} +2.73764 q^{40} +2.91604 q^{41} -11.8523 q^{42} +0.211586 q^{43} -0.216066 q^{44} -1.81156 q^{45} +2.03564 q^{46} -9.61114 q^{47} +9.26745 q^{48} +6.77411 q^{49} +1.45589 q^{50} +4.21801 q^{51} -0.811561 q^{52} +10.9331 q^{53} +3.79531 q^{54} +1.80647 q^{55} -10.1603 q^{56} +13.6394 q^{57} -7.26939 q^{58} -12.0984 q^{59} +0.262360 q^{60} +10.1997 q^{61} +2.80152 q^{62} +6.72333 q^{63} +7.46606 q^{64} +6.78527 q^{65} +5.76902 q^{66} +2.88548 q^{67} -0.229995 q^{68} -3.06702 q^{69} -5.40330 q^{70} -3.85919 q^{71} -4.95940 q^{72} +11.5899 q^{73} +3.94121 q^{74} -2.19353 q^{75} -0.743713 q^{76} -6.70445 q^{77} +21.6689 q^{78} -10.8158 q^{79} +4.22491 q^{80} -11.1529 q^{81} +4.24542 q^{82} -5.59257 q^{83} -0.973708 q^{84} +1.92293 q^{85} +0.308045 q^{86} +10.9525 q^{87} +4.94547 q^{88} -1.00000 q^{89} -2.63743 q^{90} -25.1825 q^{91} +0.167235 q^{92} -4.22094 q^{93} -13.9927 q^{94} +6.21801 q^{95} +1.48218 q^{96} +1.07064 q^{97} +9.86233 q^{98} -3.27254 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9} - q^{10} - 14 q^{11} - 3 q^{12} - 5 q^{13} - 5 q^{14} + 2 q^{15} - 3 q^{16} - 3 q^{17} + 7 q^{18} - q^{19} - 3 q^{20} - 4 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 4 q^{25} - 9 q^{26} + q^{27} + 5 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 9 q^{36} + 3 q^{37} + 2 q^{38} + 11 q^{39} + 9 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} - 9 q^{44} + 4 q^{45} - 4 q^{46} - 24 q^{47} + 21 q^{48} + q^{50} - 7 q^{51} + 8 q^{52} + 3 q^{53} + 17 q^{54} + 14 q^{55} - 13 q^{56} + 19 q^{57} + 19 q^{58} - 22 q^{59} + 3 q^{60} - 3 q^{61} - 24 q^{62} + 6 q^{63} - 11 q^{64} + 5 q^{65} + 14 q^{66} - 9 q^{67} + 31 q^{68} + 7 q^{69} + 5 q^{70} + 16 q^{71} + 10 q^{72} + 3 q^{73} + 31 q^{74} - 2 q^{75} - 24 q^{76} - 4 q^{77} + 16 q^{78} - 27 q^{79} + 3 q^{80} - 8 q^{81} - 15 q^{82} + 6 q^{83} + 7 q^{84} + 3 q^{85} + 15 q^{86} + 4 q^{87} + 30 q^{88} - 4 q^{89} - 7 q^{90} - 29 q^{91} - 17 q^{92} - 2 q^{93} + 13 q^{94} + q^{95} + 12 q^{96} + 41 q^{97} + 22 q^{98} + 21 q^{99}+O(q^{100}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9 - q^10 - 14 * q^11 - 3 * q^12 - 5 * q^13 - 5 * q^14 + 2 * q^15 - 3 * q^16 - 3 * q^17 + 7 * q^18 - q^19 - 3 * q^20 - 4 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 4 * q^25 - 9 * q^26 + q^27 + 5 * q^28 - 10 * q^29 + 6 * q^30 - 11 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 9 * q^36 + 3 * q^37 + 2 * q^38 + 11 * q^39 + 9 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 - 9 * q^44 + 4 * q^45 - 4 * q^46 - 24 * q^47 + 21 * q^48 + q^50 - 7 * q^51 + 8 * q^52 + 3 * q^53 + 17 * q^54 + 14 * q^55 - 13 * q^56 + 19 * q^57 + 19 * q^58 - 22 * q^59 + 3 * q^60 - 3 * q^61 - 24 * q^62 + 6 * q^63 - 11 * q^64 + 5 * q^65 + 14 * q^66 - 9 * q^67 + 31 * q^68 + 7 * q^69 + 5 * q^70 + 16 * q^71 + 10 * q^72 + 3 * q^73 + 31 * q^74 - 2 * q^75 - 24 * q^76 - 4 * q^77 + 16 * q^78 - 27 * q^79 + 3 * q^80 - 8 * q^81 - 15 * q^82 + 6 * q^83 + 7 * q^84 + 3 * q^85 + 15 * q^86 + 4 * q^87 + 30 * q^88 - 4 * q^89 - 7 * q^90 - 29 * q^91 - 17 * q^92 - 2 * q^93 + 13 * q^94 + q^95 + 12 * q^96 + 41 * q^97 + 22 * q^98 + 21 * q^99

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